Question

homework3: problem 29
(1 point)
if
f(x)=\frac{6 - x^{2}}{7 + x^{2}}
find ( f^{prime}(x) ).
( f^{prime}(x)= )
find ( f^{prime}(4) ).
( f^{prime}(4)= )
note: you can earn partial credit on this problem.
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Explanation:

Step1: Apply the quotient rule

The quotient rule states that if \(y = \frac{u}{v}\), then \(y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Here, \(u = 6 - x^{2}\), so \(u^\prime=-2x\); \(v = 7 + x^{2}\), so \(v^\prime = 2x\).

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Step2: Expand the numerator

Expand \((-2x)(7 + x^{2})=-14x-2x^{3}\) and \((6 - x^{2})(2x)=12x - 2x^{3}\).

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Step3: Simplify the numerator

Combine like - terms in the numerator: \(-14x-12x=-26x\), and \(-2x^{3}+2x^{3}=0\). So \(f^\prime(x)=\frac{-26x}{(7 + x^{2})^{2}}\)

Step4: Find \(f^\prime(4)\)

Substitute \(x = 4\) into \(f^\prime(x)\).

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Answer:

\(f^\prime(x)=\frac{-26x}{(7 + x^{2})^{2}}\), \(f^\prime(4)=\frac{-104}{529}\)